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DREXEL UNIVERSITY The Finite Element Method Applied to Quantum Mechanics by William Paul Czaja in the Department of Physics May 2010
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Page 1: The Finite Element Method Applied to Quantum Mechanicsbob/TermPapers/QMIII_project_Czaja.pdf · 4 PDE Tool Results and Interpretation8 ... The Finite Element Method Applied to Quantum

DREXEL UNIVERSITY

The Finite Element Method Applied to

Quantum Mechanics

by

William Paul Czaja

in the

Department of Physics

May 2010

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Contents

1 Background Theory 1

1.1 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Finite Element Method Applied to Quantum Mechanics 3

2.1 Engineers Know Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Mapping Differential Equations to Matrix Mechanics . . . . . . . . . . . . 3

2.3 Tesselation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4 Choosing the Right Basis Functions . . . . . . . . . . . . . . . . . . . . . 5

3 Tools 6

3.1 MATLAB’s Partial Differential Equation Toolbox . . . . . . . . . . . . . . 6

3.2 DistMesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 PDE Tool Results and Interpretation 8

4.1 Geometry Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Tesselation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2.1 Initial Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2.2 Refined Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3 PDE Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4 Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

i

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Chapter 1

Background Theory

1.1 Eigenvalue problems

Resonating systems display distinct patterns known has eigenmodes. Every piece of

these oscillating patterns vibrates at the same frequency λ, also known as an eigen-

value. In classical mechanical applications λ defines the normal mode frequencies, and

in quantum mechanical applications λ defines the energy levels of bound states in a po-

tential. To understand the structure of the eigenmode patterns we must take advantage

of eigenvectors. Eigenvectors maintain their direction regardless of the linear transfor-

mation the system undergoes. Examples of linear transformations include reflection,

rotation, stretching, compression, and shear. The general expression which describes

the eigenmodes is as follows:

Ax = λx (1.1)

Where A is a matrix which describes the linear transformation, x is an eigenvector, and

λ is an eigenvalue.

This can be solved using the following analytic algorithm:

1. Compute the determinant of A− λI, where I is the identity matrix.

2. Set this determinant equal to zero and solve for its roots. This is known as the

characteristic equation. Each root is an eigenvalue λ.

3. For each of these eigenvalues solve (A− λI)x = 0, where x is an eigenvector.

1

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Chapter 1. Background Theory 2

1.2 The Finite Element Method

There is a very limited set of analytic solutions that can solve eigenvalue problems.

It is exceedingly difficult to analytically examine nonlinear shapes so it’s best to use

a numerical approach. Additionaly, creating computer models of resonating systems

enables testing different initial conditions efficiently without the need of experimental

physical trials. A numerical technique known as the finite element method (FEM) is used

to make the problem more discrete by using a linear piecewise approximation. Finite

element methods solve partial differential equations for highly irregular geometries which

may require the crunching of extremely large matrices. A finite dimensional subspace

consisting of piecewise linear functions is used to approximate an infinite dimensional

problem. The FEM process begins with creating a tessellation of polygons over the

geometry. Boundary conditions are defined. Then basis functions are then chosen to

treat the piecewise approach. There are robust software packages currently available

which can handle FEMs. Software that I’ve found which can do this includes MATLAB,

ANSYS, Solid Works, and Pro-Engineer. In particular I have choosen to use MATLAB

packages to accomplish this task.

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Chapter 2

The Finite Element Method

Applied to Quantum Mechanics

2.1 Engineers Know Best

All of us are taught in our quantum mechanics courses how to analytically solve the ubiq-

uitous Schrodinger’s equation for various potentials, but our professors are not telling us

a dirty little secret. Whereas it may be beneficial to understand the underlining theory

and math behind solutions to the Schrodinger equation, the vast majority of real-world

quantum mechanical applications of the Schrodinger equation are out of our ability to

solve analytically. This is because the shapes of real-world potentials are usually highly

irregular. As responsible scientists we must sacrifice our egos and lean on methods de-

veloped by the humble yet ever-resourceful mechanical engineers. They never hesitate

to use approximate solutions if the accuracy suffices the application and it gets the job

done. Just like finite element methods can be used to solve complex resonance problems

for classically mechanical structures it can be applied to quantum mechanical problems

using the wave mechanics of the Schrodinger equation.

2.2 Mapping Differential Equations to Matrix Mechanics

The first step in applying the finite element method to quantum mechanics is to map

wave mechanics into matrix mechanics. We start by considering the variational version

of Schrodinger’s equation which expresses the wavefunction ψ of a particle, the potential

3

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Chapter 2. The Finite Element Method Applied to Quantum Mechanics 4

V it moves in, and its energy E:

δ

∫ {~2

2m∇ψ∗(x)· ∇ψ(x) +∇ψ(x) + ψ∗(x)V (x)ψ(x)− ψ∗(x)Eψ(x)

}d3x = 0 (2.1)

It is now our task to quantize the wave mechanics at a discrete set of points in con-

figuration space. To do this we define a basis set of real functions to approximate the

wavefunction and potential:

ψ∗(x) =∑i

φifi(x) V (x) =∑j

Vjfj(x) ψ(x) =∑k

ψkfk(x) (2.2)

Plugging in this basis set of functions into the variational version of Schrodinger’s equa-

tion and converting to summation convention produces our eigenvalue equation:

δ

{~2

2mφi(∇fi∇fk)ψk + φiVj(fifjfk)ψk − φiE(fifk)ψk

}= 0 (2.3)

We can interpret this now as a matrix equation:

Mikψk = 0 (2.4)

Mik = Kik + Vik − SikE (2.5)

Kik = (∇fi∇fj) (2.6)

Vik = Vj(fifjfk) (2.7)

Sik = (fifk) (2.8)

These are known as the Kinetic Energy Matrix, the Potential Energy Matrix, and the

Overlap Matrix, (K,V,S) respectively. The matrix M can be diagonalized and solved

for the eigenvalues, and eigenvectors.

2.3 Tesselation

In order to establish a linear piecewise approximation for a particular geometry it needs

to be divided into elements to create a finite dimentional subspace. This can be accom-

plished in various ways depending on the dimensions of the geometry and its irregularity.

The elements are commonly known as primitives. Some meshing algorithms segment

the edges of the geometry into equal divisions and then intelligently scatter nodes in

the subspace. The nodes are then connected to create many ploygonal primitives which

are usually choosen to be triangular. For simple configuration spaces like rectangles and

triangles the sides are evenly segmented and then the nodes of the segments are cross

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Chapter 2. The Finite Element Method Applied to Quantum Mechanics 5

connected through the subspace. As the refinment of the tesselation increases so will

the accuracy of the approximated eigenfunctions.

2.4 Choosing the Right Basis Functions

A basis on the subspace must be found to complete the discretization. Altering the

tesselation method can improve the basis. Advanced tesselation algorithms are self

adaptive and will change the placement of the nodes, the refinement of the elements, and

the order of the functions to insure the best fit possible. The basis functions (fi, fj , fk)

at a particular vertex are found through a process of unioning the basis functions of all

other elements connected to that vertex. The basis functions and their gradients are

then integrated to construct the matrix elements for the Kinetic Energy Matix K and

the Overlap Matrix S. A connectivity matrix T is used as a matrix template to fit these

elements. T describes how the coordinates of the vertices are interconnected.

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Chapter 3

Tools

3.1 MATLAB’s Partial Differential Equation Toolbox

MATLAB includes a powerful package called the Partial Differential Equation Toolbox,

PDETool for short. PDETool can be used to numerically solve PDE problems. In it

you can define 2-dimensional geometries, generate tessellations (meshes), output matri-

ces, discretize equations, produce visualizations of the normal modes, and produce an

approximate solution to the problem. The process of treating a particular problem in

PDETool can be done in two ways, through the built in graphical interface, and through

commands in a MATLAB script. PDETool uses the following partial differential equa-

tion for calculating eigenvalue problems:

−∇· (c∇u) + au = λdu (3.1)

where λ is an unknown complex eigenvalue, and a is a potential well. This problem

is treated similarly to the quantum mechanics problem given earlier. The equation is

discretized and solved algebraically. This forms the generalized eigenvalue equation:

KU = λMU (3.2)

where M is known as the mass matrix, and K is known as the stiffness matrix. This

eigenvalue equation is then run through an Arnoldi algorithm until all eigenvalues are

found on the specified interval. The Arnoldi algorithm runs through Gram-Schmidt

iterations to find orthonormal vectors of Krylov subspaces.

6

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Chapter 3. Tools 7

3.2 DistMesh

DistMesh is a mesh generator package for MATLAB. It was created by Per-Olof Pers-

son and Gilbert Strang of MIT. It can handle intelligent tesselation on very complex

geometries in 2D and 3D. The tesselation is done by using the Delaunay method.

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Chapter 4

PDE Tool Results and

Interpretation

4.1 Geometry Definition

PDETool is started by entering ’pdetool’ at the MATLAB command line. To begin an

arbitrary geometry is defined. The function pdepoly(x,y) where x and y are vectors

containing the coordinates of the vertices, can define an arbitrary geometry. You can

also draw any 2-D geometry by going to the ’Draw’ menu and selecting ’Draw Mode’

Figure 4.1: The Geometry definition for an arbitrary triangle

8

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Chapter 4. PDE Tool Results and Interpretation 9

Figure 4.2: The Geometry definition for a circular disk centered at the origin

4.2 Tesselation

4.2.1 Initial Mesh

Next we generate a mesh on the geometry. We do this by going to the ’Mesh’ menu and

selecting ’Initialize Mesh’.

Figure 4.3: The initial meshing for an arbitrary triangle

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Chapter 4. PDE Tool Results and Interpretation 10

Figure 4.4: The initial meshing for an irregular geometry

4.2.2 Refined Mesh

In order to produce a more accurate result we need to refine the mesh. We do this by

going to the ’Mesh’ menu and selecting ’Refine Mesh’. This function doubles the edge

vertices.

Figure 4.5: The refined meshing for an arbitrary triangle, refined twice

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Chapter 4. PDE Tool Results and Interpretation 11

Figure 4.6: The refined meshing for a circular disk, refined twice

Figure 4.7: The refined meshing for an circular disk with offset hole, refined twice

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Chapter 4. PDE Tool Results and Interpretation 12

Figure 4.8: The refined meshing for an irregular shape, refined twice

4.3 PDE Specification

Now that we have defined our geometry and generated a mesh we can explore the

eigenmodes. This can be achieved by hand by outputting the matrices of the mesh

to the MATLAB workspace and solving the eigenvalue problem for the eigenvectors.

PDETool can visualize the eigenmodes for you. You start by going to the ’PDE’ menu

and then selecting ’PDE Specification’. Choose ’Eigenmodes’ from this window and

then ’Ok’. With the PDE specified you can now go to the ’Solve’ menu and then select

’Solve PDE’. This will produce a visualization of the first eigenmode.

4.4 Visualizations

To view other eigenmodes go to the ’Plot’ menu and select ’Parameters’. Choose a

different eigenvalue and click ’Plot’. A three dimensional visualization of the membrane

can also be created through the plot parameters menu.

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Chapter 4. PDE Tool Results and Interpretation 13

Figure 4.9: An arbitrary triangle

Figure 4.10: An arbitrary triangle

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Chapter 4. PDE Tool Results and Interpretation 14

Figure 4.11: An arbitrary triangle

Figure 4.12: Eigenmode for an irregular shape

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Chapter 4. PDE Tool Results and Interpretation 15

Figure 4.13: Eigenmode for an irregular shape

Figure 4.14: Eigenmode for an irregular shape

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Chapter 4. PDE Tool Results and Interpretation 16

Figure 4.15: Eigenmode for a circular disk

Figure 4.16: Eigenmode for a circular disk

Figure 4.17: Eigenmode for a circular disk

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Chapter 4. PDE Tool Results and Interpretation 17

Figure 4.18: Eigenmode for a circular disk with a hole at the origin

Figure 4.19: Eigenmode for a circular disk with a hole at the origin

Figure 4.20: Eigenmode for a circular disk with a hole at the origin

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Chapter 4. PDE Tool Results and Interpretation 18

Figure 4.21: Eigenmode for a circular disk with a hole at the origin

Figure 4.22: Eigenmode for a circular disk with offset hole

Figure 4.23: Eigenmode for a circular disk with offset hole

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Chapter 4. PDE Tool Results and Interpretation 19

Figure 4.24: Eigenmode for a circular disk with offset hole

Figure 4.25: 3D eigenmode for a circular disk with offset hole

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Bibliography

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